On the variation of the energy scale 3

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1 22-Nov-15 On the variation of the energy scale 3 Page 1 On the variation of the energy scale 3 Parameters for galaxy rotation curves by Jo. Ke. Sun 22nd Nov 215

2 22-Nov-15 On the variation of the energy scale 3 Page 2 Summary The hypothesis has been put forward that the flat rotation curves of spiral galaxies arise from variations in the energy scale. A galaxy model of a Gaussian density distribution embedded in a Gaussian energy scale variation gave good fits to a small sample of six galaxies. The model requires neither dark matter nor changes to Newtonian gravitation. This paper applies the same model to a sample of 74 spiral galaxies. Reasonable fits to the rotation curves are obtained in almost all cases.

3 22-Nov-15 On the variation of the energy scale 3 Page 3 1 Introduction 1.1 The paper "On the variation of the energy scale: an alternative to dark matter" (Jo.Ke, 215) is referred to in this paper as simply "Jo.Ke 1". 1.2 The paper "On the variation of the energy scale 2: Galaxy rotation curves" (Jo.Ke, 215) is referred to in this paper as simply "Jo.Ke 2". 1.3 The rotation curves of many spiral galaxies remain flat in their outer regions and do not show the fall off in speed expected if the majority of the mass is concentrated in the galaxy centre. The widely accepted explanation for these observations is that galaxies are embedded in large haloes of dark matter. 1.4 'Jo.Ke 1' put forward the hypothesis that the flat rotation curves are caused by variations in the energy scale. The first model took a point mass galaxy and a Gaussian for the energy scale variation. This gave good fits to the outer regions of spiral galaxies. The model was applied to a small sample of just six galaxies. 1.5 'Jo.Ke 2' introduced an improved model for a spiral galaxy. This was made up of two components: (a) a narrow Gaussian density distribution, and (b) a broader Gaussian for the energy scale variation. As well as fitting the outer regions, this model also gave better fits to the inner regions. 1.6 The success of the fits to rotation curves in 'Jo.Ke 2' suggested the model should be applied to a much larger sample of spiral galaxies. This paper adopts the galaxy model of 'Jo.Ke 2' and applies it to rotation curves presented in Brownstein & Moffat (26).

4 22-Nov-15 On the variation of the energy scale 3 Page 4 4 The galaxy model 4.1 'Jo.Ke 2' modelled a spiral galaxy as an axisymmetric disk with a Gaussian density distribution embedded in a Gaussian energy scale variation. 4.2 The rotation velocity is given by where v 2 = K 2 1 Q(r) α r r Q(x) P(x) dx (1) K 2 = G M α (2) 4.3 The energy scale variation term, Q, is given by Q(r) = 1 + γ exp ( r 2 α 2 ) (3) where γ is a pure number; α is the 1/e-width of the Gaussian energy scale variation. 4.4 The density distribution term, P, is given by P(x) = 2x exp( x2 β 2 ) (4) β2 where β is the 1/e-width of the Gaussian density distribution. 4.5 The four adjustable parameters: α; γ; K; β are chosen to fit the observed rotation curves of spiral galaxies. 4.6 The Keplerian (Newtonian) rotation curve is given by setting Q(r)=Q(x)=1 in equation (1), i.e. no energy scale variation.

5 22-Nov-15 On the variation of the energy scale 3 Page 5 5 A sample of galaxy rotation curves 5.1 'Jo.Ke 1' and 'Jo.Ke 2' worked with just six galaxies. 5.2 This paper works with the galaxy rotation curves presented in Bernstein & Moffat (26). The 11 galaxies has been reduced to 74 by selecting only those where the observed rotation curves extend out to at least 1kpc. 5.3 Table (1) gives the values of the four adjustable parameters found by fitting equation (1) to the observed rotation curves. 5.4 Following 'Jo.Ke 2' an estimate for the galaxy mass is given by M = v2 α G + γ e {1 1 + γ } { 1 1 exp( α 2 β 2 ) } (5) where the velocity, v, is evaluated at the point r=α.

6 22-Nov-15 On the variation of the energy scale 3 Page 6 6 Table of parameters for galaxy rotation curves Table 1. Rotation curve parameters as derived from fitting equation (1) to the observed rotation curves for the listed galaxies. The rotation speed, v(α), is measured at the point corresponding to the characteristic distance, α. The galaxy masses, M, follow from equation (5) and are in units of 1 1 solar masses. Galaxy α kpc γ K km/s β kpc v(α) km/s F F F F IC Milky Way NGC NGC M NGC NGC NGC NGC NGC NGC NGC NGC NGC NGC NGC NGC NGC NGC

7 22-Nov-15 On the variation of the energy scale 3 Page 7 Galaxy α kpc γ K km/s β kpc v(α) km/s NGC NGC M NGC NGC NGC NGC NGC NGC NGC NGC NGC NGC NGC NGC NGC NGC NGC NGC NGC NGC NGC NGC NGC NGC NGC NGC NGC

8 22-Nov-15 On the variation of the energy scale 3 Page 8 Galaxy α kpc γ K km/s β kpc v(α) km/s NGC NGC NGC NGC NGC NGC NGC M NGC NGC NGC NGC NGC NGC NGC NGC NGC NGC NGC NGC UGC UGC UGC UGC UGC UGC

9 22-Nov-15 On the variation of the energy scale 3 Page 9 7 Figures of galaxy rotation curves The following figures show the galaxy rotation curves. The vertical axis is the speed in km/s. The horizontal axis is distance in kpc. The diamonds are the data points taken from Brownstein & Moffat (26). The solid line is an eye-fit to the data using equation (1). The dashed line is the Keplerian curve for the same mass distribution.

10 22-Nov-15 On the variation of the energy scale 3 Page 1 15 F F F F IC Milky Way NGC NGC

11 22-Nov-15 On the variation of the energy scale 3 Page NGC NGC NGC NGC NGC NGC NGC NGC

12 22-Nov-15 On the variation of the energy scale 3 Page NGC NGC NGC NGC NGC NGC NGC NGC

13 22-Nov-15 On the variation of the energy scale 3 Page NGC NGC NGC NGC NGC NGC NGC NGC

14 22-Nov-15 On the variation of the energy scale 3 Page NGC NGC NGC NGC NGC NGC NGC NGC

15 22-Nov-15 On the variation of the energy scale 3 Page NGC NGC NGC NGC NGC NGC NGC NGC

16 22-Nov-15 On the variation of the energy scale 3 Page NGC NGC NGC NGC NGC NGC NGC NGC

17 22-Nov-15 On the variation of the energy scale 3 Page NGC NGC NGC NGC NGC NGC NGC NGC

18 22-Nov-15 On the variation of the energy scale 3 Page NGC NGC NGC NGC UGC UGC UGC UGC

19 22-Nov-15 On the variation of the energy scale 3 Page UGC UGC

20 log(velocity) 22-Nov-15 On the variation of the energy scale 3 Page 2 8 Tulley-Fisher relation 8.1 The figure below plots the rotation velocity at r=α against the derived mass as set out in the table. 2.8 Mass Velocity Diagram log(mass) 8.2 A correlation is apparent. The line plotted has a slope of.3. The scatter in the data points suggests that mass varies as either the cube or fourth power of the velocity.

21 22-Nov-15 On the variation of the energy scale 3 Page 21 9 Comments 9.1 The rotation curve fits are remarkably good considering the simple nature of the model, namely a simple Gaussian density distribution and a simple Gaussian energy scale fluctuation. 9.2 Spiral galaxies are not smooth distributions of matter, but possess spiral arms and clumps of matter scattered across their disks. So it is not surprising that deviations from the fitted curves are in evidence. 9.3 In several cases the model does not fit the innermost regions of the galaxies. Some galaxies have a linear velocity curve indicating a central region with solid-body rotation. 9.4 The fits have all been done by eye. In many cases the fit is quite loose and it is possible to trade off one parameter against another. A better fitting procedure would be to carry out a grid search against a chi-squared test. 9.5 No attempt has been made to take error bars into account. In just about all cases the apparently poor fit are within the errors. 9.6 The table shows the well-known result that the Andromeda galaxy (NGC 224) is more than twice as massive as the Milky Way.

22 22-Nov-15 On the variation of the energy scale 3 Page 22 1 Conclusion 1.1 The hypothesis that variations in the energy scale can explain the rotation curves of spiral galaxies has been extended to a sample of 74 galaxies. No obvious flaws in the hypothesis have been found. 1.2 No modifications have been made to Newton's law of gravitation. No dark matter has been introduced. 1.3 A simple Gaussian density distribution for the galaxy and a simple Gaussian for the fluctuations in the energy scale go a considerable way to reproducing the observed rotation curves.

23 22-Nov-15 On the variation of the energy scale 3 Page References Brownstein, JR; Moffat, JW. (26) The Astrophysical Journal, 636, 721. Galaxy rotation curves without non-baryonic dark matter. Jo.Ke 1. (215). "On the variation of the energy scale: an alternative to dark matter". Jo.Ke 2. (215). "On the variation of the energy scale 2: galaxy rotation curves".